The CCFM uPDF evolution uPDFevolv Version 1.0.00

Abstract

uPDFevolvis an evolution code for TMD parton densities using the CCFM evolution equation. A description of the underlying theoretical model and technical realisation is given together with a detailed program description, with emphasis on parameters the user may want to change.

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PROGRAM SUMMARY Title of Program: uPDFevolv 1.0.00. Computer for which the program is designed and others on which it is operable: any with standard Fortran 77 (gfortran) and C++, tested on Linux, MAC. Programming Language used: FORTRAN 77, C++. High-speed storage required: No. Separate documentation available: No. Keywords: QCD, small x, high-energy factorization, \(k_t\)-factorization, CCFM, unintegrated PDF (uPDF), transverse momentum dependent PDF (TMD). Nature of physical problem: At high energies collisions of hadrons are described by parton densities dependent on the longitudinal momentum fraction x, the transverse momentum \(k_t\) and the evolution scale p (transverse momentum dependent (TMD) or unintegrated parton density functions (uPDF)). The evolution of the parton density with the scale p valid at both small and moderate x is given by the CCFM evolution equation. Method of solution: Since the CCFM evolution equation cannot be solved analytically, a Monte Carlo approach is applied, simulating at each step of the evolution the full four-momenta of the initial state partonic cascade. Restrictions on the complexity of the problem: None. Other Program used:Root for plotting the result. Download of the program:https://updfevolv.hepforge.org. Unusual features of the program: None.

In the framework of high-energy factorization [11, 14] the deep-inelastic scattering cross section can be written as a convolution in both longitudinal and transverse momenta of the TMD parton density function \(\mathcal{A}(x,k_t,\mu )\) with off-shell partonic matrix elements, as follows

The factorization formula (1) allows one to resum logarithmically enhanced \( x \rightarrow 0 \) contributions to all orders in perturbation theory, both in the hard scattering coefficients and in the parton evolution, taking fully into account the dependence on the factorization scale \(p\) and on the factorization scheme [24, 25].

The CCFM evolution equation [21, 22, 23] is an exclusive equation for final state partons and includes finite-\(x\) contributions to parton splitting. It incorporates soft gluon coherence for any value of \(x\).

1.1.1 Gluon distribution

The evolution equation for the TMD gluon density \({\mathcal {A}}(x,k_t,p)\), depending on \(x\), \(k_t\) and the evolution variable \(p\), is

where \(\Delta _s\) is the Sudakov form factor, and \(\mathcal{A}_0 ( x , k_t, q_0 )\) is the starting distribution at scale \(q_0\). The integral term in the right hand side of Eq. (2) gives the \(k_t\)-dependent branchings in terms of the Sudakov form factor \(\Delta _s\) and unintegrated splitting function \(P\).

In addition to the full splitting function, simplified versions are useful in applications and are made available. One uses only the singular parts of the splitting function (set by Ipgg=0, ns=0 in uPDFevolv):

In general a four-momentum a can be written in light-cone variables as a = \((a^+, a^-, a_T)\) with \(a^+ \) and \(a^-\) being the light-cone components and \(a_T\) being the transverse component. The CCFM (as well as the BFKL) evolution depends only on one of the light-cone components. Assuming that the other one can be neglected, this leads to the condition that the virtuality of the parton propagator \(a^2 = 2 a^+ a^- -a_T^2\) should be dominated by the transverse component, while the contribution from the longitudinal components is required to be small. The condition that \(a^+a^- = 0\) leads to the so-called consistency constraint (see Fig. 1), which has been implemented in different forms (set by Ikincut=1,2,3 in uPDFevolv)

In Eqs. (15), (16) the non-Sudakov form factor is not included, unlike the CCFM kernel given in the appendix B of [22], because we only associate this factor with \(1/z\) terms. The term \(x{ Q_v}_0\) in Eq. (15) is the contribution of the non-resolvable branchings between starting scale \(q_0\) and evolution scale \(p\), given by

with \({\tilde{q}} = q - zk_t\), and \(q (k_t)\) being the transverse momentum of the quark (gluon).

The evolution equation for the TMD sea-quark density \(\mathcal{S}(x,k_t,p)\), depending on \(x\), \(k_t\) and the evolution variable \(p\) is (we allow a general \(k_t\) dependence of the splitting functions, as proposed in appendix B of [22], even if it is not included in Eqs. (18)–(21)),

In a Monte Carlo (MC) solution [34, 35] we evolve from \(q_0\) to a value \(q'\) obtained from the Sudakov factor \(\Delta _s(q',q_0)\) (for a schematic visualisation of the evolution see Fig. 2). Note that the Sudakov factor \(\Delta _s(q',q_0)\) gives the probability for evolving from \(q_0\) to \(q'\) without resolvable branching. The value \(q'\) is obtained from solving for \(q'\):

If \(q' > p\) then the scale \(p\) is reached and the evolution is stopped, and we are left with just the first term without any resolvable branching. If \(q'<p\) then we generate a branching at \(q'\) according to the splitting function \(\tilde{P}(z') \), as described below, and continue the evolution using the Sudakov factor \(\Delta _s(q'',q')\). If \(q'' > p\) the evolution is stopped and we are left with just one resolvable branching at \(q'\). If \(q''<p\) we continue the evolution as described above. This procedure is repeated until we generate \(q>p\). By this procedure we sum all kinematically allowed contributions in the series \(\sum f_i(x,p)\) and obtain an MC estimate of the parton distribution function.

1.2 Computational techniques: CCFM grid

When using the CCFM evolution in a fit program to determine the starting distribution \(\mathcal{A}_0 (x)\), a full MC solution [34, 35] is no longer suitable, since it is time consuming and suffers from numerical fluctuations. Instead a convolution method introduced in [37, 38] is used. The kernel \( \tilde{\mathcal{A}}(x'',k_t,p) \) is determined once from the Monte Carlo solution of the CCFM evolution equation, and then folded with the non-perturbative starting distribution \(\mathcal{A}_0 (x)\),

The kernel \(\tilde{\mathcal{A}}\) incorporates all of the dynamics of the evolution, including Sudakov form factors and splitting functions. It is determined on a grid of \(50\otimes 50\otimes 50\) bins in \( x, k_t, p\). The binning in the grid is logarithmic, except for the longitudinal variable \(x\) where we use 40 bins in logarithmic spacing below 0.1, and 10 bins in linear spacing above 0.1.

Using this method, the complete coupled evolution of gluon and sea quarks is more complicated, since it is no longer a simple convolution of the kernel with the starting distribution. To simplify the approach, here we allow only for one species of partons at the starting scale, either gluons or sea-quarks. During evolution the other species will be generated. This approach, while convenient for QCD fits, has the feature that sea-quarks, in the case of gluons only at \(q_0\), are generated with perturbative transverse momenta (\(k_t> k_{t\ cut}\)), without contribution from the soft (non-perturbative) region.

1.3 Functional forms for starting distribution

1.3.1 Standard parametrisation

For the starting distribution \(\mathcal{A}_0\), at the starting scale \(q_0\), the following form is used:

with \(R_0^2(x) = (x/x_0)^\lambda \). The free parameters are \(\sigma _0 = A_2\), \(\lambda = A_3\), \(x_0 = A_4\) and \(\alpha _s=A_5\). In order to be able to use this type of parameterisation over the full \(x\) range, an additional factor of \((1-x)^{A_6}\) (see [40]) is applied.

1.4 Plotting TMDs

A simple plot program is included in the package. For a graphical web interface use TMDplotter [41].

1.5 Application

The evolution of the TMD gluon density has been used to perform fits to the DIS precision data [42, 43], as described in detail in [38].

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